A unified finite element algorithm for two-equation models of turbulence

Abstract This paper presents a simple change of dependent variables for turbulence quantities that achieves two goals: the resulting formulation preserves positivity of all turbulence variables; and leads naturally to a simple algorithm applicable to all two-equation models of turbulence. It is also a useful means for comparing the structure of different turbulence models. The approach consists in solving for the natural logarithm of the turbulence variables. The methodology is illustrated by applying it to three popular models: the standard k – ϵ model; the k – τ model of Speziale; and the k – ω model of Wilcox. When logarithmic variables are used, transport equations for one model are obtained by adding or subtracting those from another model and by using the simple relationships that exist between the logarithms of k , ϵ , ω and τ . An existing adaptive finite element algorithm developed for the logarithmic form of the k – ϵ model is applied to the other models without any change. The formulation is verified on a shear layer possessing a closed form solution. The approach is then applied to turbulent flow over backward facing step for which measurements are available. Computations show that solutions of controlled accuracy can be achieved using the same solution algorithm for all models thus opening the way to systematic comparison studies.